Moon Jae-in says North Korea wants to speed up the agreed denuclearisation process and arrange a new meeting with Trump.

North Korean leader Kim Jong-un has requested a second summit with US President Donald Trump and also wants US Secretary of State Mike Pompeo to visit Pyongyang soon, according to South Korea's president.

Speaking at a news conference upon his return to Seoul on Thursday, Moon Jae-in said Kim told him during his three-day summit in Pyongyang that he wanted to hasten the process of denuclearisation.

"Kim Jong-un expressed his wish that he wanted to complete denuclearisation quickly and focus on economic development," Moon said.

"He said he hoped Mike Pompeo would visit North Korea soon, and also a second summit with Trump would take place in the near future, in order to move the denuclearisation process along quickly."

He also said he would carry a private message from Kim for Trump when he meets the US president in New York City next week at the UN General Assembly.

Moon will also convey to Trump both his and Kim's desire to obtain a declaration ending the Korean War by the end of this year, he said.

Asked about the specific corresponding US measures demanded by Kim for denuclearisation, Moon said the matter should be discussed between North Korea and the US.

"Pyongyang joint declaration did not include any of those specific measures of North Korea and the corresponding actions of the US. We discussed some of it, but we did not have it written down so I cannot reveal any of the discussed items with you here," he said.

Singapore summit

Kim and Trump held a landmark and high-profile meeting in Singapore in June, where the North's leader committed to work towards denuclearisation of the Korean Peninsula, but no details were agreed.

Washington and Pyongyang have subsequently sparred over what that means and how it will be achieved, with the Trump administration consistently referring to the denuclearisation of North Korea specifically.

Experts were sceptical but the Trump administration immediately welcomed the move, with Pompeo inviting his North Korean counterpart to meet next week on the sidelines of the UN General Assembly.

Moon said on Thursday that Kim would visit Seoul around November, no later than the end of the year.

He also said Kim agreed to have a meeting between two national assemblies and to facilitate cooperation of the municipal governments.

"I also asked for North Korea to release its restrictions on Kumgangsan facilities in December. I am going to host a cultural event or exhibition with North Korea to celebrate 110th year of Korea," Moon said.